On Finite Semigroup Cross-Sections and Complete Rewriting Systems Antonio´ Malheiro Centro de Algebra´ da Universidade de Lisboa Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal and Faculdade de Cienciasˆ e Tecnologia da UNL, Quinta da Torre, 2829-516 Monte de Caparica, Portugal E-mail: [email protected] Abstract solvable word problem is invariant for any finite presenta- tion defining the same semigroup. Thus we can refer to the In this paper we obtain a [finite] complete rewriting sys- word problem of a given semigroup and not only to some tem defining a semigroup/monoid S, from a given finite of its finite presentations. right cross-section of a subsemigroup/submonoid defined Finite and complete (that is, noetherian and confluent) by a [finite] complete presentation. In the semigroup case rewriting systems are used to solve word problems among the subsemigroup must have a right identity element which other algebraic decision problems (see [3, 8, 11] for exam- must also be part of the cross-section. In the monoid case ples). The word problem is solved using the ‘normal form the submonoid and the cross-section must include the iden- algorithm’: given two words u and v, we can calculate irre- tity of the semigroup. The result on semigroups allow us ∗ ∗ ducible elements u0 and v0 such that u→R u0 and v→R v0, to show that if G is a group defined by a [finite] com- ∗ and we conclude that u↔R v if and only if u0 and v0 are plete rewriting system then the completely simple semi- identical words. This application reveals the importance of group M[G;I,J;P] is also defined by a [finite] complete such rewriting systems. rewriting system. In Group Theory we say that a subgroup H has finite in- dex in a group G if there are finitely many cosets Hg, for any g ∈ G. In that case we get G as a disjoint union of fi- 1. Introduction nitely many cosets H, Hg1, ..., Hgn. The set {1,g1,...,gn} is said to be a cross-section for H in G. This means that One of the most important common topics of Computer each element in G is uniquely representable as the product Science and Mathematics is the word problem. This prob- of an element of H by an element of {1,g1,...,gn}. Groves lem was initially formulated in the group theory by M. and Smith [5] proved that if H is a subgroup of finite index Dehn in 1912 [4]. However, it can be introduced in a nat- in G and H has a [finite] complete rewriting system then G ural way in the semigroup theory. A semigroup presenta- has a [finite] complete rewriting system. The main goal of tion P is a pair hX | Ri where X is an alphabet and R is this paper is to extend this result first for the monoid case a rewriting system on X, that is, a binary relation in the and second to the semigroup case. free semigroup on X. Each semigroup presentation ori- Let S be a semigroup and let M be a subsemigroup of gins a semigroup S(P) resultant of the quotient of the free S. A [finite] subset T is said to be a [finite] (right) cross- + ∗ section for M in S if any element of S can be uniquely fac- semigroup X by the Thue congruence ↔R generated by R. This congruence is the symmetric, reflexive and tran- torized in the form mt, with m ∈ M and t ∈ T. + The first result can be stated without any other extra con- sitive closure of the binary relation →R on X given by: ∗ dition and as we will see the proof can be depicted from the for any (r+1,r−1) ∈ R and any words w1 and w2 on X , we group case. have w1r+1w2 →R w1r−1w2. Now, the word problem can + be formulated as follows: given words u and v in X , de- Theorem 1.1 Let S be a monoid with identity element 1S. cide whether or not u and v represent the same element of Let M be a submonoid of S and let T be finite right cross- ∗ S(P), that is, decide if u↔R v. If there exists an algorithm section for M in S. Suppose that 1S ∈ M ∩T. If M is defined which solves the word problem for any two words then the by a [finite] complete rewriting system, then S is also de- word problem is said to be solvable. The property of having fined by a [finite] complete rewriting system. Our second result requires an extra condition on the sub- 2. Preliminaries semigroup. We say the an element e of a semigroup M is a right identity element if me = m, for any m ∈ M. Let X be an alphabet. We denote by X∗ the free monoid on X and by X+, the free semigroup on X. For an element Theorem 1.2 Let S be a semigroup and let M be a sub- of X∗, a word w, we denote the length of w by |w|. semigroup of S having a right identity element e. Suppose A rewriting system is a pair (X,R), where X is an al- that there is a finite right cross-section for M in S includ- phabet and R is a binary relation in X∗. The elements of ing the element e. If M is defined by a [finite] complete R are referred to as rewriting rules. Usually, a rewriting rewriting system, then S is also defined by a [finite] com- rule r ∈ R is written in the form r = (r+1,r−1) or, simply, plete rewriting system. r+1 → r−1. In the following the rewriting system will be simply denoted by R. We say that a rewriting system R is This last result has a very important consequence in finite if both R and X are finite. Semigroup Theory. A non-empty set A of a semigroup In X∗ we define a binary relation, → , denoted as single- S is called a left ideal if SA ⊆ A, a right ideal if AS ⊆ A R step reduction, in the following way: and an ideal if it is both a left and a right ideal. Com- pletely simple semigroups, among other characterizations u→ v ⇔ u = w1r+1w2 and v = w1r−1w2 (see [6]), are those semigroups with no proper ideals and R ∗ with minimal left and right ideals. for some (r+1,r−1) ∈ R and w1,w2 ∈ X . The transitive In the 1920’s Suschkewitsch [12] has obtained a descrip- and reflexive closure of → is denoted by −→∗ . By −→+ R R R tion of the completely simple semigroups in the following ∗ we denote the transitive closure of →R . A word u ∈ X terms: let G be a group, let I and J be non-empty sets and is said to be R-reducible, if there is a word v ∈ X∗ such let P = (p ji) be a J × I matrix with entries in G; the set that u→R v. If a word is not R-reducible, it is called R- I × G × J with multiplication irreducible or simply irreducible. By Irr(R) we denote the set of all R-irreducible words. (i1,s1, j1)(i2,s2, j2) = (i1,s1 p j1i2 s2, j2), ∗ By ↔R we denote the equivalence relation induced by → which is a congruence on the free monoid X∗. The is a completely simple semigroup; conversely, every com- R ∗ pletely simple semigroup is isomorphic to a semigroup quotient of the free monoid X by what is called the Thue ∗ constructed in this way. The semigroup constructed as congruence ↔R generated by R gives the monoid defined above is denoted by M[G;I,J;P] and it is called the I × J by R and it is denoted by M(X;R). The set X is called the Rees matrix semigroup over the group G with the sandwich generating set and R the set of defining relations. More matrix P. generally, a monoid is said to be defined by the rewriting This description will allow us to use the above theorem system R if M =∼ M(X;R). Thus, the elements of M are and conclude the following about completely simple semi- identified with congruence classes of words from X∗. We groups: will sometimes identify words and elements they represent. We say that a rewriting system R on X is noetherian if Theorem 1.3 Let G be a group, let I and J be non-empty the relation →R is well-founded, in other words, if there is finite sets and let P = (p ji) be a J ×I matrix with entries in no infinite descending chains G. If the group G is defined by a [finite] complete rewriting system then the completely simple semigroup M[G;I,J;P] w1 →R w2 →R w3 →R ···→R wn →R ··· . is also defined by a [finite] complete rewriting system. ∗ We say that R is confluent if whenever we have u−→R v ∗ 0 ∗ ∗ To understand the finiteness condition on the sets I and and u−→R v there is a word w ∈ X such that v−→R w and 0 ∗ J on the above result it is essential to have in mind the re- v −→R w. If R is simultaneously noetherian and confluent sult [2] by H. Ayik and N. Ruskuc,ˇ where the authors con- we say that R is complete.